MNRAS 000, 1–15 ()
Preprint September 1, 2016
Compiled using MNRAS LATEX style file v3.0
Milankovitch Cycles of Terrestrial Planets in Binary Star
Systems
Duncan
Forgan
1
1?
arXiv:1608.05592v2 [astro-ph.EP] 31 Aug 2016
Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy, University of St Andrews, North Haugh, KY16 9SS
Accepted
ABSTRACT
The habitability of planets in binary star systems depends not only on the radiation
environment created by the two stars, but also on the perturbations to planetary orbits and rotation produced by the gravitational field of the binary and neighbouring
planets. Habitable planets in binaries may therefore experience significant perturbations in orbit and spin. The direct effects of orbital resonances and secular evolution
on the climate of binary planets remain largely unconsidered.
We present latitudinal energy balance modelling of exoplanet climates with direct
coupling to an N Body integrator and an obliquity evolution model. This allows us
to simultaneously investigate the thermal and dynamical evolution of planets orbiting binary stars, and discover gravito-climatic oscillations on dynamical and secular
timescales.
We investigate the Kepler-47 and Alpha Centauri systems as archetypes of P and S
type binary systems respectively. In the first case, Earthlike planets would experience
rapid Milankovitch cycles (of order 1000 years) in eccentricity, obliquity and precession,
inducing temperature oscillations of similar periods (modulated by other planets in
the system). These secular temperature variations have amplitudes similar to those
induced on the much shorter timescale of the binary period.
In the Alpha Centauri system, the influence of the secondary produces eccentricity variations on 15,000 year timescales. This produces climate oscillations of similar
strength to the variation on the orbital timescale of the binary. Phase drifts between
eccentricity and obliquity oscillations creates further cycles that are of order 100,000
years in duration, which are further modulated by neighbouring planets.
Key words:
astrobiology, methods:numerical, planets and satellites: general
1
INTRODUCTION
Approximately half of all solar type stars reside in binary
systems (Duquennoy & Mayor 1991; Raghavan et al. 2010).
Recent exoplanet detections have shown that planet formation in these systems is possible. Planets can orbit one of
the stars in the so-called S type configuration, such as γ
Cephei (Hatzes et al. 2003) HD41004b (Zucker et al. 2004)
and GJ86b (Queloz et al. 2000). If the binary semimajor axis
is sufficiently small, then the planet can orbit the system
centre of mass in the circumbinary or P type configuration.
Planets in this configuration were first detected around postmain sequence stars, in particular the binary pulsar B16026 (Thorsett et al. 1993; Sigurdsson et al. 2003). The Ke-
?
E-mail:[email protected]
c The Authors
pler space telescope has been pivotal in detecting circumbinary planets orbiting main sequence stars, such as Kepler-16
(Doyle et al. 2011), Kepler-34 and Kepler-35 (Welsh et al.
2012), and Kepler-47 (Orosz et al. 2012).
Planets in binary systems are sufficiently common that
we should consider their habitability seriously. As of July
2016, 112 exoplanets have been detected in binary star systems1 , giving an occurrence rate of around 4% (previous estimates on a much smaller exoplanet population by Desidera
& Barbieri 2007 placed the fraction of planets in S type systems at 20%). At gas giant masses, the occurrence rate of
planets around P type binaries is thought to be similar to
that of single stars (Armstrong et al. 2014b).
1
http://www.univie.ac.at/adg/schwarz/multiple.html
2
Duncan Forgan
However, theoretical modelling indicates that the dynamical landscape of the binary significantly affects the
planet formation process, both for S-type (Wiegert & Holman 1997; Quintana et al. 2002, 2007; Thébault et al. 2008,
2009; Xie et al. 2010; Rafikov & Silsbee 2014b,a) and P-type
systems (Doolin & Blundell 2011; Rafikov 2013; G. Martin
et al. 2013; Marzari et al. 2013; Dunhill & Alexander 2013;
Meschiari 2014; Silsbee & Rafikov 2015). Therefore, when
considering the prospects for habitable worlds in the Milky
Way, one must take care to consider the effects that companion stars will have on the thermal and gravitational evolution
of planets and moons.
The habitable zone (HZ) concept (Huang 1959; Hart
1979) is often employed to determine whether a detected
exoplanet might be expected to be conducive to surface liquid water (that is, if its mass and atmospheric composition
allow it). Initially calculated for the single star case using
1D radiative transfer modelling of the layers of an Earthlike
atmosphere (Kasting et al. 1993), this quickly establishes a
range of orbital distances that produce clement planetary
conditions. Over time, line radiative transfer models have
been refined, leading to improved estimates of the inner and
outer habitable zone edges (Kopparapu et al. 2013, 2014).
In the case of multiple star systems, the presence and
motion of extra sources of gravity and radiation have two
important effects:
(i) The morphology and location of the system’s HZ
changes with time, and
(ii) Regions of the system are orbitally unstable
These joint thermal-dynamical constraints on habitability
have been addressed in a largely decoupled fashion using a
variety of analytical and numerical techniques.
The thermal time dependence of the HZ can be evaluated by combining the flux from both stars, taking care
to weight each contribution appropriately, and applying the
single star constraints to determine whether a particular spatial location would receive flux conducive to surface water.
Kane & Hinkel (2013) use the aggregate flux to find a peak
wavelength of emission. Assuming the combined spectrum
resembles a blackbody, Wien’s Law provides an effective
temperature for the total insolation, and hence the limits
of Kopparapu et al. (2013) can be applied. This approximation is acceptable for P type systems, where the distance
from each star to the planet is similar.
Haghighipour & Kaltenegger (2013) and Kaltenegger &
Haghighipour (2013) weight each star’s flux by its effective
temperature, and then determine the regions at which this
weighted flux matches that of a 1 M star at the habitable
zone boundaries. This approach is suitable for both S type
and P type systems. A detailed analytic solution for calculations of this nature has been undertaken by Cuntz (2014).
Mason et al. (2013) take a similar approach, but they
also note that for P type systems, the tidal interaction between primary and secondary can induce rotational synchronisation, which can reduce extreme UV flux and stellar wind
pressure, improving conditions in the habitable zone compared to the single star case (see also Zuluaga et al. 2016).
The dynamical constraints on habitability rely heavily
on N Body simulation, most prominently the work of Dvorak (Dvorak 1984, 1986) and Holman & Wiegert (1999). By
integrating an ensemble of test particles in a variety of orbits
around a binary, regions of dynamical instability can be determined. Holman & Wiegert (1999) used these simulations
to develop empirical expressions for a critical orbital semimajor axis, ac . In the case of a P type system, this represents
a minimum value - anything inside ac is orbitally unstable,
as given by the following expression:
ap > ac = abin ((1.6 ± 0.04) + (5.1 ± 0.05)ebin
+(4.12 ± 0.09)µ − (2.22 ± 0.11)e2bin − (4.27 ± 0.17)µebin
−(5.09 ± 0.11)µ2 + (4.61 ± 0.36)µ2 e2bin . (1)
In the case of an S-type system, ac represents a maximum
value:
ap < ac = abin ((0.464 ± 0.006) − (0.38 ± 0.01)µ
−(0.631 ± 0.034)ebin + (0.586 ± 0.061)µebin
+(0.15 ± 0.041)e2bin − (0.198 ± 0.074)µe2bin
(2)
where abin is the binary semimajor axis, ebin is the binary
orbital eccentricity, and µ represents the binary mass ratio:
µ=
M2
M1 + M2
(3)
The majority of binary habitability calculations rely on
the above dynamical constraints. Notable exceptions include
Eggl et al. (2012)’s use of Fast Lyapunov Indicators for chaos
detection, which yield slightly smaller values of ac for S type
systems (Pilat-Lohinger & Dvorak 2002), and Jaime et al.
(2014)’s use of invariant loops to discover non-intersecting
orbits (Pichardo et al. 2005). There is a good deal of research into spin-orbit alignments of extrasolar planets under
the influence of inclined stellar companions (e.g. Anderson
et al. 2016), but this work rarely pertains to terrestrial planet
habitability. On the other hand, the evolution of planetary
rotation period has been studied intently with regards to
habitability of planets in single star systems (e.g. Bolmont
et al. 2014; Brown et al. 2014; Cuartas-Restrepo et al. 2016).
All the above approaches to determining habitability in
binary systems rely on an initial 1D calculation of the atmosphere’s response to radiative flux, where the key dimension
is atmospheric depth. Equally, 1D approaches can consider
the latitudinal variation of flux on a planet’s surface, giving rise to the so-called latitudinal energy balance models or
LEBMs, which have been used both in the single star case
(Spiegel et al. 2008; Dressing et al. 2010; Vladilo et al. 2013)
and for multiple stars (Forgan 2012, 2014). These are better
suited to capture processes that depend on atmospheric circulation, such as the snowball effect arising from ice-albedo
feedback (Pierrehumbert 2005; Tajika 2008), which is likely
to occur in systems where the orbits undergo Milankovitch
cycles and other secular evolution (Spiegel et al. 2010).
However, all these approaches typically decouple the
thermal from the dynamical. The orbital constraints on the
HZ are considered separately from the radiative transfer
calculations. While they are eventually combined, the binary habitable zones that are constructed do not incorporate
the effects of coupled gravito-thermal perturbations. Indeed,
Holman & Wiegert (1999) admit that their empirical limits
on semimajor axis ignore the potential for stable resonances
MNRAS 000, 1–15 ()
Milankovitch Cycles in Binary Systems
inside the instability region, as well as unstable resonances in
stable regions (cf Chavez et al. 2014). It is likely that planets
on stable orbits in binary systems will experience relatively
strong orbital element evolution. For example, circumbinary
planets can undergo rapid precession of periapsis, which affects their ability to be detected via transit (Kostov et al.
2014; Welsh et al. 2015). Presumably the spin evolution of
planets in this situation can proceed with similar rapidity.
Crucially, climate systems are nonlinear, and can alter their
state on very short timescales compared to the planet’s orbital period.
In this work, we consider coupled gravito-thermal perturbations on the climate of exoplanets in binary systems. To
do so, we present a LEBM directly coupled to an N-Body integrator and an obliquity evolution model. We use this combined code to investigate the spin-orbital-climate dynamics
of putative planets in two archetypal binary systems: the
P-type system Kepler-47, a multi-planet circumbinary system which possesses one exoplanet inside the habitable zone
(Orosz et al. 2012); and Alpha Centauri, the nearest star system to the Sun, an S type binary system which was thought
to possess a short period, Earth-mass exoplanet (Dumusque
et al. 2012)2 . By evolving the orbits of the bodies in tandem
with the climate, we are able to detect climate variations
that are directly linked to the binary’s orbit, and the secular evolution of the planet’s orbit and spin.
In section 2, we describe the LEBM, and how the
N Body model is coupled to it. In section 3 we describe
the simulation setup and results on dynamical and secular
timescales, in section 4 we discuss the implications for habitability, and in section 5 we summarise the work.
2
Latitudinal Energy Balance Modelling
Typically, LEBMs solve the following diffusion equation:
C
∂T (x, t) ∂
−
∂t
∂x
D(1 − x2 )
∂T (x, t)
∂x
2014). In our approach, we consider a given latitude to be
habitable if its temperature resides within 273K < T < 373
K, i.e. that surface water is liquid.
The diffusion coefficient D determines the efficiency of
heat redistribution across latitudes. Its value is defined such
that a fiducial Earthlike planet, rotating with period 1 day,
orbiting at 1 au around a star of 1M , produces the correct average temperature profile (see e.g. Spiegel et al. 2008;
Vladilo et al. 2013). If the planet’s rotation is more rapid,
the Coriolis effect will inhibit latitudinal heat transport (see
Farrell 1990):
D = 5.394 × 102
Ωrot
Ωrot,⊕
−2
,
(5)
where Ωrot is the rotational angular velocity of the planet,
and Ωrot,⊕ is the rotational angular velocity of the Earth.
This is a necessarily simple expression, but can be made
more rigorous through including terms for atmospheric pressure and mean molecular weight (e.g. Williams & Kasting
1997, but see also Vladilo et al. 2013’s attempts to introduce a latitudinal dependence to D to mimic the Hadley
convective cells on Earth). Beyond this, full global circulation modelling is needed to explore the effects of rotation
(Del Genio 1993, 1996).
As in previous work, we solve equation (4) using an explicit forward time, centre space finite difference algorithm.
A global timestep is used, with standard constraint
∆tLEBM <
(∆x)2 C
.
2D(1 − x2 )
(6)
The atmospheric heat capacity C, is a function of the
planet’s surface ocean fraction and how much of that is
frozen, fice :
METHOD
2.1
3
= S(1−A(T ))−I(T ).
C = fland Cland + focean ((1 − fice )Cocean + fice Cice ) ,
(7)
where fland = 1 − focean . The heat capacities of land, ocean
and ice covered areas are
(4)
Where T (x, t) is the surface temperature, C is the effective
heat capacity of the atmosphere, S is the insolation flux, I
is the IR cooling and A is the albedo. In the above equation,
C, S, I and A are functions of x (either explicitly, as S is, or
implicitly through T ). The latitude λ appears through x ≡
sin λ. This equation is evolved with the boundary condition
dT
= 0 at the poles (where λ = [−90, 90]◦ ), and requires
dx
the assumption that the planet rotates rapidly relative to
its orbital period. Our implementation of the LEBM follows
that of Spiegel et al. (2008), and has been used previously in
studying the climate evolution of planets in binary systems
on timescales of order a few hundred years (Forgan 2012,
Cland = 5.25 × 109 erg cm−2 K−1
Cocean = 40.0Cland
9.2Cland
Cice =
2Cland
The infrared cooling function I is
I(T ) =
σSB T 4
,
1 + 0.75τIR (T )
τIR (T ) = 0.79
This detection is no longer considered to be credible by some
groups, due to concerns with how stellar activity is filtered out
of radial velocity data (Hatzes 2013). Recent attempts to detect
α Cen Bb via transit show a null result (Demory et al. 2015),
and re-analysis of the radial velocity data suggests that α Cen
Bb does not exist (Rajpaul et al. 2016).
MNRAS 000, 1–15 ()
(8)
with the optical depth of the atmosphere given as
2
263 K < T < 273 K
T < 263 K.
T
273 K
3
.
(9)
The albedo function is
A(T ) = 0.525 − 0.245 tanh
T − 268 K
.
5K
(10)
4
Duncan Forgan
This correctly reproduces the ice-albedo feedback phenomenon, which allows a rapid non-linear increase in albedo
as the ice coverage increases.
At any instant, for a single star, the insolation received
at a given latitude at an orbital distance r is
S = q0 cos Z
1AU
r
,
∆ti =
M
M
4
erg s−1 cm−2
(12)
S = q0 µ̄.
(14)
We do this by integrating µ over the sunlit part of the day,
i.e. h = [−H, +H], where H(x) is the radian half-day length
at a given latitude. Multiplying by H/π (as H = π if a
latitude is illuminated for a full rotation) gives the total
diurnal insolation as
H
π
µ̄ =
q0
(H sin λ sin δ + cos λ cos δ sin H) . (15)
π
+
+
ji
si
si
ji
!1/2
.
(17)
Here, a represents the magnitude of the body’s acceleration,
ji si and ci are the magnitudes of the first, second and third
derivatives of the acceleration of particle i respectively, and η
is a tunable parameter which we set to 0.002. This is a fairly
strict timestep condition, and as such the error in angular
momentum is typically one in 106 or better throughout.
2.3
Obliquity Evolution
We adopt the obliquity evolution model of Laskar (1986a,b),
developed for the Solar System and subsequently used for
putative exoplanet systems (Armstrong et al. 2004, 2014a).
In this paradigm, the evolution of the obliquity δ0 and precession pa are functions of the inclination variables
i
sin Ω
2
i
q = sin
cos Ω
2
p = sin
(18)
(19)
Where i is the inclination, and Ω is the longitude of the ascending node. The obliquity and precession evolve according
to the following:
dδ0
= −B sin pa + A cos pa
dt
dpa
= R(δ0 ) − cot δ0 (A sin pa + B cos pa ) − 2C − pg .
dt
(20)
(21)
A, B and C are all functions of p and q:
The radian half day length is calculated as
cos H = − tan λ tan δ.
η
ai
ji
ci
si
(13)
The solar hour angle is h, and δ is the solar declination,
which is calculated by computing the scalar product of the
spin-axis vector s and the planet-star separation vector r.
We obtain the spin-axis vector by rotation of the angular
momentum vector in the x-axis by δ0 , followed by a rotation
around the axis defined by the angular momentum vector
by pa , the axial precession angle (or longitude of winter solstice).
Our rapid rotation assumption requires that we use diurnally averaged quantities, so we also diurnally average S:
The dynamical evolution of the system utilises a standard 4th-order Hermite integrator with an adaptive shared
timestep. We calculate this N Body timestep for all
bodies{i}, ∆tN , by finding the minimum value of ∆ti :
(11)
cos Z = µ = sin λ sin δ + cos λ cos δ cos h.
S = q0
The N-Body Model
2
where q0 is the bolometric flux received from the star at a
distance of 1 AU, and Z is the zenith angle:
q0 = 1.36 × 106
2.2
(16)
The total insolation is a simple linear combination of the
contributions from both stars. If one star is eclipsed by the
other, then we set its contribution to S to zero. We ensure that the simulation can accurately model an eclipse by
adding an extra timestep criterion, ensuring that the transit’s duration will not be less than ten timesteps.
We fix the parameters of the model to those of the
Earth: the initial obliquity is set to 23.5 degrees, and the
ocean fraction focean = 0.7. The rotation period of the body
is 1 day. It is important to note that altering these parameters will alter the strength of climate fluctuations, especially
if orbits are eccentric. Indeed, Forgan (2012) showed that
reducing the planet’s ocean fraction can significantly boost
temperature fluctuations in S-type binary systems with fixed
orbits, and that increasing obliquity while holding other parameters fixed typically increases the average temperature of
the planet. The following results should be considered with
these facts in mind.
2
A(p, q) = p
(q̇ − pC(p, q))
1 − p2 − q 2
2
B(p, q) = p
(ṗ − qC(p, q))
1 − p2 − q 2
C(p, q) = ṗq − q̇p
(22)
(23)
(24)
Note that these A, B, C terms ensure increases in inclination mediate changes in obliquity. Equivalently, if the
inclination of a planet’s orbit is increased, the obliquity decreases, as the angle between the orbital plane and the fundamental plane defined by the planet’s spin axis decreases
(see Figure 1 of Armstrong et al. 2014a).
That being said, the spin axis of the planet can change
regardless of the inclination, due to either direct torques
from the star (R(δ0 )) or from the relativistic precession term
pg . Laskar (1986a) give the direct torque from a single host
star as
R(δ0 ) =
3k2 M∗
ED S0 cos δ0
a3 Ωrot
(25)
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Milankovitch Cycles in Binary Systems
Where ED is the dynamical ellipticity (i.e. the nonsphericity) of the planet (which we set equal to 0.00328005
for the remainder of this work),
S0 =
−3/2
1
1 − e2
− 0.422 × 10−6
2
(26)
∗
and k2 = GM
(where the units of G must be selected
4π 2
to be appropriate for comparison with Ωrot ). For a single
star, the relativistic precession is
pg =
kr
2(1 − e2 )
(27)
where
kr =
c2
n3 a2
(1 + Mp /M )
k2
(1 + Mp /M )
(29)
In this work, we make the following assumptions about these
equations in their use for binary stars. In the S type case,
we assume that direct torques and precession is generated by
the host star only. The secondary can influence the obliquity
only through modification of the planet’s orbital elements
e, i, Ω.
In the case of a P type system, we assume that the
torques from both stars co-add. The planet’s orbital elements relative to the system centre of mass are employed in
both cases for simplicity. Given the distance of both stars
from the centre of mass is small relative to the planet’s semimajor axis, this seems a reasonable assumption (although we
do note the need for further investigation of this problem,
see Discussion).
2.4
Coupling the Models
To couple the LEBM to the N Body integrator and obliquity evolution model, we elect the simplest route, by forcing all systems to evolve according to a shared timestep.
In practice, this means comparing the LEBM and N Body
timesteps, i.e.
∆t = min (∆tN , ∆tLEBM ) .
(30)
Typically the obliquity evolution timestep is much larger
than the other two. This does limit the code’s efficacy when
evolving systems with either short dynamical timescales, or
short thermal timescales. In the case of a fiducial Earth-Sun
model, we are able to evolve the coupled LEBM-N Body system with similar runtime to a LEBM using fixed Keplerian
orbits. We will see that in the S type configuration, the addition of N Body physics makes little appreciable difference
to computational speed. However, in the P type configuration, the short dynamical timescale of the binary increases
the runtime significantly. This could be alleviated by other
timestepping approaches, which we address in the Discussion.
MNRAS 000, 1–15 ()
We emphasise that correctly resolving the LEBM is crucial - it is a nonlinear system, with positive feedback mechanisms that can operate rapidly compared to the system’s
spin-orbit dynamical time. It is this property that requires
the models to be fully coupled in order to truly understand
the climate of planets in dynamically rich systems over secular timescales.
We have tested the N Body integrator and obliquity evolution model against the results of Armstrong et al. (2014a)
(their System 1), and find a good match for their orbital elements and spin parameters. In a companion paper (Forgan
and Mead, in prep) we test the spin-orbit-climate evolution
of the Earth under the influence of the Solar System planets,
and find that appropriate Milankovitch cycles in the planet’s
spin-orbit parameters do indeed arise.
(28)
The mean motion n can be determined by considering k in
the context of Kepler’s third law:
n2 a3 =
5
3
RESULTS
We now apply our combined model to the two archetypal P
and S type binary systems. We will be comparing runs with
obliquity evolution switched on and off to investigate what
climate features are due to either orbital or spin evolution.
3.1
3.1.1
Kepler-47
Setup
The Kepler-47 system contains a 1.043 M star and an 0.362
M star orbiting each other with a period of around 7.5
days. We adopt the orbital parameters of Orosz et al. (2012),
with a semi-major axis of 0.0836 AU and eccentricity 0.0234,
and assume that the stars’ luminosities are determined by
standard main sequence relations.
Kepler-47c orbits inside the circumbinary habitable
zone at 0.989 AU, with an eccentricity upper limit of 0.41. As
we are using the Kepler-47 system as an archetype for terrestrial habitability in P type systems, we replace Kepler-47c
with an Earth mass planet orbiting at the same semi-major
axis, and investigate both low and high eccentricity orbits.
Kepler-47b orbits interior to Kepler-47c with a semimajor
axis of 0.2956 AU with eccentricity 0.034, and period 49.5
days. We investigate the climate of our terrestrial planet
both with and without Kepler-47b’s presence.
3.1.2
Zero Eccentricity, Without Kepler-47b
Figure 1 shows the orbital evolution of a terrestrial planet
orbiting the Kepler-47 binary at ap = 0.989 AU with zero
eccentricity and an initial inclination of 0.5◦ relative to the
binary plane. We run the simulation for 10,000 years, with
sufficiently high snapshot frequency that the orbital period
of the binary (0.0205 years) is well resolved. The planet’s
orbit is relatively stable, undergoing small eccentricity and
inclination variations of around 800 and 400 year periods
respectively (note also that the argument of periapsis precesses on a similar timescale).
In the case where the obliquity is fixed, the planet’s
climate settles to a stable state, with mean temperatures
fluctuating by around 0.1 K (top row of Figure 2). We can
see in the periodogram for fixed obliquity that the major
contribution to temperature fluctuation is seasonal variation
6
Duncan Forgan
Figure 1. The dynamical evolution of the terrestrial planet with Kepler-47c’s semimajor axis, and zero eccentricity. Left, the orbital
evolution of the body, as given by its eccentricity and inclination. Right, the spin evolution as given by the obliquity and precession
angles.
Figure 2. The climate evolution of the Kepler-47c terrestrial planet, with obliquity evolution switched off (top row) and switched on
(bottom row). Left: The global maximum, minimum and mean temperatures on the surface over 10,000 years. Right: Periodograms for
the mean temperature. The red dashed lines indicate the planet’s orbital period of 0.829 years, and its harmonics (1/2, 1/3... of the
period).
MNRAS 000, 1–15 ()
Milankovitch Cycles in Binary Systems
over the orbital period of 0.829 years (and its harmonics
at 1/n of the period), closely followed by a contribution at
the binary period of 0.0205 years as the relative insolation
from each object varies. Finally, we see a significantly weaker
contribution from eccentricity variation at 800 years. There
are no low order mean motion resonances between the binary
and planet period - the system is closest to a 80:2 resonance.
There is no evidence of such a resonance in the temperature
data, which would result in a peak at approximately 1.66
years in the periodogram.
In the case where obliquity is allowed to vary (bottom
row of Figure 2), we can immediately detect climatic variations from inspecting the maximum, mean and minimum
temperature curves. The presence of an extra peak at around
400 years in the temperature periodogram (bottom right of
Figure 2) shows that the inclination is forcing similar variations in obliquity and precession angle (Figure 1). Generally
speaking, the planet’s climate now shows a richer set of resonant features in the periodogram with periods greater than
that of the orbital periods in play.
not in mean motion resonance, the contribution of the binary
to the planet’s eccentricity periodogram is smeared between
0.02 and 0.03 years due to the planet’s increased eccentricity. Note that this increased eccentricity raises the maximum temperature beyond the runaway greenhouse limit of
340K. The runaway greenhouse effect is not modelled by the
LEBM, and we should be careful when making statements
about this configuration’s habitability. Some weak modes
appear around the planet’s orbital period of 0.829 years,
but their origin is unclear - presumably they are linked to
the precession of the planet’s periapsis relative to that of the
binary.
Allowing obliquity to vary allows other oscillations to
assume greater importance. Indeed, the variations caused by
binary motion are close to negligible in this case, especially
compared to variations in the year-decade range.
3.2
3.2.1
3.1.3
Zero Eccentricity, with Kepler-47b
The previous section has shown that single planets in P type
systems will undergo secular evolution quite similar to that
of Milankovitch cycles (albeit at a much reduced timescale).
We now add Kepler-47b to the system (with zero eccentricity and inclination) to gauge what effect neighbouring
planets might have on the secular evolution of circumbinary
habitable climates.
Figure 3 shows the orbital evolution of the Kepler-47c substitute. Comparing to the previous section (Figure 1), we see
that the eccentricity variation has not changed much, but
the inclination variation has decreased its period by a factor
of roughly two. Interestingly, no such changes are seen in
the obliquity and precession evolution, indicating that stellar torques are presumably dominant.
The periodograms for both cases (Figure 4) show little
change in the climate by adding a neighbour planet. The
periodograms show no signs of Kepler-47b’s influence at its
orbital period of 0.1355 years. The features seen at 0.1355
years with obliquity evolution exist in the previous run without Kepler-47b. The planets are not in mean motion resonance - they are closest to a 49:8 mean motion resonance,
which would indicate a peak at approximately 6.63 years,
which is not seen in either case.
3.1.4
High Eccentricity, no b
We now remove Kepler-47b from the system, and increase
the eccentricity of our habitable planet to 0.4. The dynamical evolution (Figure 5) is more rapid, with small eccentricity and inclination oscillations about the original value
with a period of around 550 years, and similar obliquity and
precession evolution. Note the amplitude modulation of the
inclination, which coincides with peak eccentricity.
Naturally, the climate of the body experiences stronger
temperature oscillations even with obliquity switched off
(top row of Figure 6). The periodogram shows greater importance for the seasonal variation, as well as the eccentricity
variation peak at 550 years. As the planet and binary are
MNRAS 000, 1–15 ()
7
Alpha Centauri B
Setup
The Alpha Centauri system is in fact a hierarchical triple
system, with Alpha Centauri A and B orbiting each other at
23.4 AU with eccentricity 0.5179. We neglect the third component, Proxima Centauri, as it orbits at great distance and
is of sufficiently low mass (Wertheimer & Laughlin 2006).
We consider α Cen B as the host star for a planetary system.
The stellar masses are MA = 1.1M , MB = 0.934M ,
and their luminosities are LA = 1.519L and LB = 0.5L
respectively (Thevenin et al. 2002). This modifies the location of the habitable zone as was previously measured by
Forgan (2012), as they used main sequence relations for the
luminosity.
We do not model the presence of α Cen Bb, as its 3
day orbit would place it extremely close to α Cen B, and
hence is unlikely to produce a significant perturbation on
any planets within the habitable zone. Instead, we place a
single Earthlike planet in the system near the outer edge of
the habitable zone, on a circular orbit at 0.7095 AU , where
the effects of α Cen A are maximal. To ensure obliquity
evolution occurs, we give our planet a small inclination of
0.5◦ relative to the binary plane.
However, we do wish to consider the relative strength
of Milankovitch cycles resulting from the binary compared
to those induced by neighbouring planets (cf Figure 8 of
Andrade-Ines & Michtchenko 2014). We attempt to maximise this effect by running another set of models with a
second Earth-mass body orbiting in 3:2 resonance with our
habitable world (with a zero inclination orbit).
3.2.2
Single Planet Runs
Figure 7 shows the dynamical evolution of the planet around
α Cen B. The initially zero eccentricity is forced to a maximum of 0.05 on a cycle of approximately 14,500 years. The
obliquity and precession evolve with a slightly longer period,
resulting in the eccentricity and obliquity cycles drifting in
and out of phase.
This phase drift results in markedly different climate
evolution of the body, compared to the case where obliquity
8
Duncan Forgan
Figure 3. The dynamical evolution of the terrestrial planet with Kepler-47c’s semimajor axis, and zero eccentricity, in the presence of
Kepler-47b. Left, the orbital evolution of the body, as given by its eccentricity and inclination. Right, the spin evolution as given by the
obliquity and precession angles.
Figure 4. The climate evolution of the Kepler-47c terrestrial planet in the presence of Kepler-47b, with obliquity evolution switched
off (top row) and switched on (bottom row). Left: The global maximum, minimum and mean temperatures on the surface over 10,000
years. Right: Periodograms for the mean temperature. The red dashed lines indicate Kepler-47b’s orbital period of 0.1355 years, and its
harmonics (1/2, 1/3... of the period).
MNRAS 000, 1–15 ()
Milankovitch Cycles in Binary Systems
9
Figure 5. The dynamical evolution of the terrestrial planet with Kepler-47c’s semimajor axis, and eccentricity 0.4. Left, the orbital
evolution of the body, as given by its eccentricity and inclination. Right, the spin evolution as given by the obliquity and precession
angles.
Figure 6. The climate evolution of the Kepler-47c terrestrial planet at high eccentricity, with obliquity evolution switched off (top row)
and switched on (bottom row). Left: The global maximum, minimum and mean temperatures on the surface over 10,000 years. Right:
Periodograms for the mean temperature. The red dashed lines indicate the planet’s orbital period of 0.829 years, and its harmonics (1/2,
1/3... of the period).
MNRAS 000, 1–15 ()
10
Duncan Forgan
Figure 7. The dynamical evolution of the terrestrial planet orbiting α Cen B. Left, the orbital evolution of the body, as given by its
eccentricity. We refrain from plotting the inclination, as its fluctuations are extremely low with no obvious periodic oscillation. Right,
the spin evolution as given by the obliquity and precession angles.
Figure 8. The climate evolution of the α Cen B terrestrial planet, with obliquity evolution switched off (top row) and switched on
(bottom row). Left: The global maximum, minimum and mean temperatures on the surface over 100,000 years (obliquity evolution off)
and over approximately 300,000 years (obliquity evolution on). Right: Periodograms for the mean temperature. The red dashed lines
indicate the binary’s orbital period of 79 years, and its harmonics (1/2, 1/3... of the period).
MNRAS 000, 1–15 ()
Milankovitch Cycles in Binary Systems
is held fixed (Figure 8). In the fixed obliquity case, the eccentricity cycle induces a temperature oscillation of approximately 2K (to add to the radiative oscillation of 5K due to
the changing proximity of α Cen A). The periodogram shows
the two dominant oscillation modes at 79.9 and 14,500 years.
Their strength is indicated by the strength of their subsequent harmonics, which can be seen down to the tenth level!
A quite different picture emerges if obliquity evolution is
activated (bottom row of Figure 8). The temperature oscillations are now modulated by the phase drift between eccentricity and obliquity, which is periodic over ∼ 200,000 year
timescales. When the two cycles are in phase, we see the
largest temperature oscillations (e.g. at t ∼ 200,000 years).
bits and stronger Milankovitch cycling. Given that planet
formation models disfavour the creation of Jupiter mass bodies in this system (Xie et al. 2010) and are ruled out by
observations of the α Cen system, at least at periods less
than ∼ 1 year (Endl et al. 2001; Dumusque et al. 2012; Demory et al. 2015) this is not a particular concern. But, one
might imagine that undetected Neptune mass bodies could
be present in this system on relatively long period orbits,
and such bodies would be responsible for longer period Milankovitch cycles similar to that of Earth’s.
4
3.2.3
Adding a planet in 3:2 mean motion resonance
We now consider joint planetary-binary Milankovitch cycles
by adding an Earth mass planet on a circular orbit at 0.9293
AU, placing it in 3:2 mean motion resonance with the habitable planet. Test runs with α Cen A absent show the additional planet induces regular eccentricity oscillations in the
habitable planet with amplitude of approximately 0.01, and
a period of approximately 500 years. Incidentally, the absence of α Cen A would also place both planets outside the
habitable zone.
With α Cen A present, the combination of stellar and
planetary forcings produces eccentricity oscillations of maximum amplitude 0.08 (left panel of Figure 9) and with a mix
of dominant periods, as opposed to the distinct 14,500 year
period observed in the single planet case. The inclination
varies with a period of approximately 30,000 years, with a
distinctive shift in mean inclination of around 0.001 radians
(i.e. 0.05◦ ). The obliquity and precession continue to evolve
at close to the eccentricity oscillation period, but the amplitude of their oscillations varies on approximately twice this
timescale.
The uniform temperature evolution cycles seen in Figure 8 are now more confused with the addition of a neighbour planet (Figure 10). With obliquity evolution switched
off (top row), the extra structure introduced into the eccentricity and inclination oscillations leaves an imprint on the
temperature curves. This can be seen in its periodogram (top
right panel of Figure 10), which shows a relatively weak feature at the perturbing planet’s orbital period, and at the resonant period of twice the perturber’s period (or equivalently,
three times the habitable planet’s period). The perturbations induced by the additional planet produce temperature
variations of up to 2K compared to the single planet case.
With obliquity evolution turned on (bottom panel), the
eccentricity/obliquity relationship seen in the previous case
is preserved, resulting in phase drift between the two oscillations. However, the extra structure in the eccentricity oscillation prevents the smooth amplitude modulation of temperature that we saw in the bottom right panel of Figure 8. It is
broadly present, but heavily modified by the presence of the
neighbouring planet. The periodogram still reveals weak signals at the perturbing planet’s period, and the strong peak
feature at approximately 14,500 years is now split in two.
There is also a significant increase in signal for periods of
order 100-1000 years.
Additional giant planets in a system like this might be
expected to produce even larger excursions from circular orMNRAS 000, 1–15 ()
11
4.1
DISCUSSION
Limitations of the Model
LEBM modelling is by its definition a compromise between
the granularity of a climate simulation and computational
expediency. This compromise is stretched further by the coupling of the N-Body integrator and obliquity evolution. We
have adopted a very simple coupling where both the N-Body
and LEBM components are constrained to follow the same
global timestep.
This timestep system works extremely well for systems
where the dynamical timescale is relatively long, such as the
S type binary systems. In this scenario, the system timestep
is limited only by the LEBM, and as such we can run simulations with similar wallclock times as that of a LEBM using
fixed Keplerian orbits. However, in the P type scenario, the
dynamical timescale is relatively short, and the system is
limited by the N Body timestep required to resolve the binary.
There are several possible strategies for mitigating this
timestep issue. The most straightforward solution is to adopt
a non-shared timestep for the N-Body component, allowing
some of the bodies to possess shorter N Body timesteps. This
would reduce the computational load of evolving all the bodies (and the LEBM) at what can be very short timesteps.
Another solution would require the interpolation of body
motions (in the case where the LEBM timestep is small
compared to the N Body timestep), but this would likely
produce only marginal gains in speed. Perhaps the best solution for P type systems would be chain regularisation of
the tight binary orbit (Mikkola & Aarseth 1990, 1993).
Aside from the new challenges arising from the adoption
of the N-Body integrator, there are the usual limitations
that many LEBMs are subject to. Our implementation of
the LEBM is among the most simple available which can
still broadly reproduce the seasonal temperature profiles of
a fiducial Earth model. The principal advantage of this simplicity is its ease of interpretation, but we must acknowledge
that more advanced models may produce features we cannot.
For example, we do not model the carbonate-silicate
(CS) cycle, which moderates fluctuations in atmospheric
temperature by increasing and reducing the partial pressure
of carbon dioxide. The timescale on which we expect CO2
levels to vary depends on the planet’s geochemical properties, especially its ocean circulation. For Earthlike planets,
the equilbriation timescale of CO2 is approximately half a
million years (Williams & Kasting 1997) which is far shorter
than the Milankovitch cycles experienced by the planetary
12
Duncan Forgan
Figure 9. The dynamical evolution of the terrestrial planet orbiting α Cen B. Left, the orbital evolution of the body, as given by its
eccentricity and inclination. Right, the spin evolution as given by the obliquity and precession angles.
Figure 10. The climate evolution of the α Cen B terrestrial planet, with obliquity evolution switched off (top row) and switched on
(bottom row). Left: The global maximum, minimum and mean temperatures on the surface over 100,000 years (obliquity evolution off)
and over approximately 300,000 years (obliquity evolution on). Right: Periodograms for the mean temperature. The red dashed lines
indicate the binary’s orbital period of 79 years, and its harmonics (1/2, 1/3... of the period).
MNRAS 000, 1–15 ()
Milankovitch Cycles in Binary Systems
bodies in this analysis. However, our understanding of the
CS cycle is rooted firmly in our understanding of the Earth,
which orbits a single star. It remains unclear whether a
planet in a binary star system would possess a similar equilibriation timescale, even if the planet was effectively identical to the Earth.
While we have taken the first steps towards coupling
celestial dynamics and LEBM climate modelling here, there
are still several steps ahead of us. For example, the tidal
interactions between bodies will also modify orbits of habitable worlds, in particular reducing their eccentricity and
modifying their rotational period (Bolmont et al. 2014;
Cunha et al. 2014). While this is unlikely to be an issue
for the orbital configurations adopted in this analysis, it remains the case that while the tidal interactions between the
binary stars is well characterised (e.g. Mason et al. 2013;
Zuluaga et al. 2016), the tidal evolution of planets in P type
systems remains relatively unexplored.
Also to be explored in full are the obliquity variations
felt by planets in binary systems. We have adopted a set of
equations designed for a single star planetary system, and
assumed they are valid when there are two stars present.
In effect we have assumed that in S type systems, the secondary’s direct tidal torque on planetary spin is negligible,
and that in P type systems the direct torques always co-add.
Is this always the case? More investigation is needed.
We should also note that the strength of Milankovitch
cycles measured by the LEBM will be an underestimate.
Tests conducted using Solar system parameters (Forgan &
Mead, in prep.) give Milankovitch cycles for the Earth that
are an order of magnitude smaller in temperature variation than observed in paleoclimate data (Zachos et al. 2001;
Lisiecki & Raymo 2005). Paradoxically, stochastic EBMs,
with additional random noise, can enhance periodic variations through the phenomenon of stochastic resonance
(Imkeller 2001; Benzi 2010). Obliquity variation does produce a much richer set of temperature variations on decadal
timescales, which may be forced into stochastic resonant behaviour under appropriate circumstances. Future investigations should consider adding a random noise term to the
LEBM equation to permit this behaviour.
4.2
Implications for Habitability
So what have we gained by this coupling of N Body and
LEBM integrators? Initially, we are able to confirm that in
general, the decoupled approach of considering the radiative
and gravitational perturbations separately is broadly acceptable.
Previous work in this field is not invalidated by our results, but it makes explicit some general principles that are
already known implicitly. Firstly, the habitable zone of a
planetary system is defined by more than where the radiation sources are in the system. The gravitational sources are
equally important. We know this on Earth thanks to our
understanding of Milankovitch cycles, and the Earth’s orbital and spin cycles are relatively weak when compared to
measured cycles for Earthlike planets in typical exoplanet
system configurations around a single star (Spiegel et al.
2010, Forgan & Mead, in prep.).
Secondly, the habitable zone of binary systems is even
more sensitive to the gravitational field than single star sysMNRAS 000, 1–15 ()
13
tems. This is already demonstrated implicitly by the N-Body
simulations of orbital stability discussed in the Introduction. Our results clearly identify the effect of orbital and
spin stability on climate. We show that relatively strong
Milankovitch cycles exist in binary systems, even if there
is only one planet present. The periods of these cycles are
in general shorter than that of single star systems, but of
similar amplitudes. Even on short timescales, the radiative
perturbations induced over the orbital period of the binary
are detectable in the mean temperature of the planet.
Thirdly, the circadian rhythms of life on planets in binary systems will be forced to adapt to the rhythms present
in the binary system, as is evidenced by analogous studies
of lunar photoperiodism in terrestrial organisms (O’MalleyJames et al. 2012; Forgan et al. 2015 and references within).
Temperature fluctuations of several K on timescales ranging from less than a year to almost a century (depending
on whether the system is P or S type) is likely to produce significant fluctuations in surface coverage of biomes.
The rapid Milankovitch cycles are likely to play a stronger
role also. More sophisticated climate models coupled to NBody physics (for example, 3D global circulation models)
may show potential for more, shorter Ice Ages, and briefer
interglacial periods. The presence of such rapid changes to
environmental selection pressure will have an indelible effect
on the evolution of organisms in binary planetary systems.
Future work should build on recent attempts to produce 3D
General Circulation Models of circumbinary planets (cf May
& Rauscher 2016), incorporating the systems’ gravitational
evolution to determine these effects in detail.
5
CONCLUSIONS
We have investigated Milankovitch cycles both circumbinary
(P type) and distant binary (S type) systems, using Kepler47 and α Centauri as archetypes. To do this, we coupled a
1D latitudinal energy balance climate model (LEBM) with
an N-Body integrator to follow the orbital evolution, and an
obliquity evolution algorithm to study the spin-axis evolution.
We find that the combined spin-orbit-radiative perturbations induced by a companion star on a habitable planet
produce Milankovitch cycles for both types of binary system, even when other planets are not present. Periodogram
analysis identifies both dynamical and secular oscillations
in the mean temperature of planets in these systems, over
a variety of short and long periods, as well as the presence
of radiative perturbations directly linked to the period of
the binary. The strength of these oscillations is sensitive to
the orbital configuration of the system. The relative phase
between eccentricity, precession and obliquity cycles is important, just as it is for the Earth.
In general, we find these Milankovitch cycles are significantly shorter than comparable cycles on the Earth (in
some cases shorter than 1000 years), although the amplitude
of the changes they produce in the planets’ orbital elements
are comparable to those experienced by Earth. This work
demonstrates the need to consider joint dynamics-climate
simulations of habitable worlds in binary systems, if we are
to truly assess the potential for the birth and growth of biospheres on worlds with two suns.
14
Duncan Forgan
ACKNOWLEDGMENTS
DHF gratefully acknowledges support from the ECOGAL
project, grant agreement 291227, funded by the European
Research Council under ERC-2011-ADG. This work relied
on the compute resources of the St Andrews MHD Cluster. The author thanks both Nader Haghighipour and James
Gilmore for insightful comments on an early version of this
manuscript. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. The code used
in this work is now available open source as OBERON, which
can be downloaded at github.com/dh4gan/oberon.
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